Cheap serverless containers using API Gateway
Sometimes I need to run a long-lived app. In those cases I reach for AWS ECS Fargate instead of AWS Lambda. You can run a container on Fargate for as little as $9/month, or $2.70/month if you're happy to roll the dice with Fargate Spot (I usually do!)
If you have a web app, you almost certainly use a load balancer in front of your containers. And this is where the cost goes from "fun side project" to "oh, I'm not sure I'm willing to spend that much money on this." The load balancer by itself is at least $16.40/month - you could run six containers for that price!